• No results found

Superconvergence of semidiscrete finite element methods for bilinear parabolic optimal control problems

N/A
N/A
Protected

Academic year: 2020

Share "Superconvergence of semidiscrete finite element methods for bilinear parabolic optimal control problems"

Copied!
14
0
0

Loading.... (view fulltext now)

Full text

(1)

R E S E A R C H

Open Access

Superconvergence of semidiscrete finite

element methods for bilinear parabolic

optimal control problems

Yuelong Tang

1,2*

and Yuchun Hua

1

*Correspondence:

[email protected] 1Institute of Computational

Mathematics, College of Science, Hunan University of Science and Engineering, Yongzhou, Hunan 425100, China

2Office of Continuing Education,

Peking University, Beijing, 100871, China

Abstract

In this paper, a semidiscrete finite element method for solving bilinear parabolic optimal control problems is considered. Firstly, we present a finite element approximation of the model problem. Secondly, we bring in some important intermediate variables and their error estimates. Thirdly, we derive a priori error estimates of the approximation scheme. Finally, we obtain the superconvergence between the semidiscrete finite element solutions and projections of the exact solutions. A numerical example is presented to verify our theoretical results.

MSC: 49J20; 65M60

Keywords: superconvergence; finite element method; bilinear parabolic optimal control problems

1 Introduction

We consider the following bilinear parabolic optimal control problem:

min

uK

 

T

y(t,x) –yd(t,x)L()+u(t,x)

L()

dt, (.)

∂ty(t,x) –div

A(x)∇y(t,x)+u(t,x)y(t,x) =f(t,x), tJ,x, (.)

y(t,x) = , tJ,x, (.)

y(,x) =y(x), x, (.)

where∈Ris a convex polygon with the boundary, andJ= [,T] ( <T< +∞). The

coefficient matrixA(x) = (aij(x))×∈[W,∞(¯)]×is a symmetric and positive definite.

Moreover, we assume thatf(t,x)∈C(J;L()),y(x)∈H(), and the admissible control

setKis defined by

K=v(t,x)∈LJ;L():av(t,x)≤b, a.e. inJ×,

where ≤a<bare real numbers.

There has been a wide range of research on finite element approximation of elliptic op-timal control problems. For finite element solving linear and semilinear elliptic control problems, a priori error estimates were investigated in [] and [], and superconvergence

(2)

were established in [] and [], respectively. Yang et al. [] obtained the superconvergence of finite element approximation of bilinear elliptic control problems. In addition, some similar results of mixed finite element approximation for linear elliptic control problems can be found in [, ].

In recent years, there are a lot of related works on finite element approximation of parabolic optimal control problems, mostly focused on linear or semilinear cases. A pri-ori error estimates of space-time finite element and standard finite element approxima-tion for linear parabolic control problem were derived in [] and []. The superconver-gence of variational discretization and standard finite element approximation for semilin-ear parabolic control problem can be found in [] and [], respectively.

As far as we know, there has been little work done on bilinear parabolic control prob-lems. In this paper, we purpose to obtain the superconvergence properties of semidiscrete finite element method for bilinear parabolic optimal control problems.

We adopt the notation Wm,q() for Sobolev spaces on with norm ·

Wm,q() and

seminorm | · |Wm,q(). We set H

()≡ {vH() :v| = } and denote Wm,() by

Hm(). We denote byLs(J;Wm,q()) the Banach space of allLsintegrable functions from

J into Wm,q() with norm v

Ls(J;Wm,q())= (T

vsWm,q()dt)

s for s[,∞) and the

standard modification for s=∞. Similarly, we can define the spaceHl(J;Wm,q()) and

Ck(J;Wm,q()) (see e.g. []). In addition, letcorCbe generic positive constants.

The rest of this paper is organized as follows. A semidiscrete finite element approxima-tion of (.)-(.) is presented in Secapproxima-tion . Some important intermediate variables and their error estimates are introduced in Section . In Section , a priori error estimates of the approximation scheme are derived. In Section , the superconvergence between pro-jections of the exact solutions and the finite element solutions is obtained. A numerical example is presented to illustrate our theoretical results in the last section.

2 A semidiscrete finite element approximation

We now consider a standard semidiscrete finite element approximation of (.)-(.). To ease the exposition, we denoteLp(J;Wm,q()) and · Lp(J;Wm,q())by Lp(Wm,q) and

· Lp(Wm,q)respectively. LetW =H() andU=L(). Moreover, we denote · Hm()

and · L()by · mand · , respectively. Let

a(v,w) =

(Av)· ∇w, ∀v,wW,

(f,f) =

f·f, ∀f,f∈U.

From the assumptions onAwe have

a(v,v)≥cv

, a(v,w) ≤Cvw, ∀v,wW.

The weak formulation of (.)-(.) can be read as follows:

min

uK

 

T

yyd+u

dt, (.)

(∂ty,w) +a(y,w) + (uy,w) = (f,w), ∀wW,tJ, (.)

(3)

It follows from (see e.g. []) that problem (.)-(.) has at least one solution (y,u) and that if the pair (y,u)∈(H(L)L(H

))×K is a solution of (.)-(.), then there is a

costatep∈(H(L)L(H

)) such that the triplet (y,p,u) meets the following optimality

conditions:

(∂ty,w) +a(y,w) + (uy,w) = (f,w), ∀wW,tJ, (.)

y(,x) =y(x), ∀x, (.)

–(∂tp,q) +a(q,p) + (up,q) = (yyd,q), ∀qW,tJ, (.)

p(T,x) = , ∀x, (.)

T

(uyp,vu)dt≥, ∀vK. (.)

As in [], it is easy to get the following lemma.

Lemma . Let(y,p,u)be the solution of(.)-(.).Then

u=minmax(a,yp),b. (.)

LetP be the space of polynomials not exceeding , andTh be regular triangulations

ofsuch that¯ =τT¯andh=maxτTh{}, where denotes the diameter of the

elementτ. Furthermore, we set

Uh=

vhL() :vh|τ= constant,∀τTh

,

Wh=

vhC(¯) :vh|τ∈P,∀τTh,vh|= 

.

As in [], we assume that

Kh=

vhUh:avh|τb,∀τTh

is a closed convex set inUh. We recast a semidiscrete finite element approximation of

(.)-(.) as

min

uhL(Kh)

 

T

yhyd+uh

dt, (.)

(∂tyh,wh) +a(yh,wh) + (uhyh,wh) = (f,wh), ∀whWh,tJ, (.)

yh(,x) =yh(x), ∀x, (.)

whereyh(x) =Rh(y(x)), andRhis an elliptic projection operator, which will be specified

later.

It is well known that (.)-(.) again has a solution (yh,uh) and that if the pair (yh,uh)∈

H(W

hL(Kh) is a solution of (.)-(.), then there is a costatephH(Wh) such that

the triplet (yh,ph,uh) meets the following conditions:

(∂tyh,wh) +a(yh,wh) + (uhyh,wh) = (f,wh), ∀whWh,tJ, (.)

(4)

–(∂tph,qh) +a(qh,ph) + (uhph,qh) = (yhyd,qh), ∀qhWh,tJ, (.)

ph(T,x) = , ∀x, (.)

T

(uhyhph,vhuh)dt≥, ∀vhKh. (.)

We introduce the averaging operatorπc

hfromUontoUhas

πhcv|τ=

 |τ|

τ

v dx, ∀τTh, (.)

where|τ|is the measure ofτ. Then we can similarly derive the following lemma.

Lemma . Let(yh,ph,uh)be the solution of(.)-(.).Then we have

uh=min

maxa,πhc(yhph)

,b. (.)

3 Error estimates of intermediate variables

In this section, we introduce some important intermediate variables and derive some re-lated error estimates. For allvK, lety(v),p(v)∈H(L)∩L(H) satisfy the following equations:

∂ty(v),w

+ay(v),w+vy(v),w= (f,w), ∀wW,tJ, (.)

y(v)(,x) =y(x), ∀x, (.)

∂tp(v),q

+aq,p(v)+vp(v),q=y(v) –yd,q

, ∀qW,tJ, (.)

p(v)(T,x) = , ∀x. (.)

Letyh(v),ph(v) meet the following system:

∂tyh(v),wh

+ayh(v),wh

+vyh(v),wh

= (f,wh), ∀whWh,tJ, (.)

yh(v)(,x) =yh(x), ∀x, (.)

∂tph(v),qh

+aqh,ph(v)

+vph(v),qh

=yh(v) –yd,qh

, ∀qhWh,tJ, (.)

ph(v)(T,x) = , ∀x. (.)

If (y,p,u) and (yh,ph,uh) are the solutions of (.)-(.) and (.)-(.), respectively, then

(y,p) = (y(u),p(u)) and (yh,ph) = (yh(uh),ph(uh)).

We define an elliptic projection operatorRh:WWhthat satisfies

a(Rhφφ,wh) = , ∀φW,whWh, (.)

and theL-orthogonal projection operatorQ

h:UUhthat satisfies

(5)

They have the following properties (see e.g. []):

RhφφsCh–, ∀φH(),s= , , (.)

QhψψsCh+s|ψ|, ∀ψH(),s= , . (.)

The following lemmas are very important for a priori error estimates and superconver-gence analysis.

Lemma . For any vK,if there exists a constant c> such that

cwa(w,w) + (vw,w), ∀wW, (.)

then (.)-(.) and (.)-(.) have unique solutions, respectively. Assuming that y(v),p(v)∈H(H),we have

y(v) –yh(v)L(L)+y(v) –yh(v)L(H)Ch, (.)

p(v) –ph(v)L(L)+p(v) –ph(v)L(H)Ch. (.)

Proof It follows from (.)-(.) and (.)-(.) that

∂t

y(v) –yh(v)

,wh

+ay(v) –yh(v),wh

+vy(v) –yh(v)

,wh

= ,

whWh,tJ, (.)

y(v)(,x) –yh(v)(,x) =y(x) –yh(x), ∀x, (.)

∂t

p(v) –ph(v)

,qh

+aqh,p(v) –ph(v)

+vp(v) –ph(v)

,qh

=y(v) –yh(v),qh

, ∀qhWh,tJ, (.)

p(v)(T,x) –ph(v)(T,x) = , ∀x, (.)

Lettingwh=Rhy(v) –yh(v) in (.), we obtain, for anytJ,

 =∂t

y(v) –yh(v)

,Rhy(v) –yh(v)

+ay(v) –yh(v),Rhy(v) –yh(v)

+vy(v) –yh(v)

,Rhy(v) –yh(v)

. (.)

Applying (.) to (.), we have

 

d

dty(v) –yh(v) 

+cy(v) –yh(v)

∂t

y(v) –yh(v)

,y(v) –yh(v)

+ay(v) –yh(v),y(v) –yh(v)

+vy(v) –yh(v)

,y(v) –yh(v)

=∂t

y(v) –yh(v)

,y(v) –Rhy(v)

+ay(v) –yh(v),y(v) –Rhy(v)

+vy(v) –yh(v)

,y(v) –Rhy(v)

. (.)

From (.) and (.), Hölder’s inequality, Young’s inequality with, and Gronwall’s

(6)

Lemma . For anyυ,ωK,let(y(υ),p(υ))and(y(ω),p(ω))be the solutions of(.)-(.), (yh(υ),ph(υ)),and let(yh(ω),ph(ω))be the solutions of(.)-(.).Then

y(υ) –y(ω)L(L)+y(υ) –y(ω)L(H)ωL(H–), (.)

p(υ) –p(ω)L(L)+p(υ) –p(ω)L(H)ωL(H–), (.)

yh(υ) –yh(ω)L(L)+yh(υ) –yh(ω)L(H)ωL(H–), (.)

ph(υ) –ph(ω)L(L)+ph(υ) –ph(ω)L(H)ωL(H–). (.)

Proof For anyυ,ωK, it is clear that

∂t

y(υ) –y(ω),w+ay(υ) –y(ω),w+υy(υ) –y(ω),w

=ωυ,wy(ω), ∀wW,tJ, (.)

y(υ)(,x) –y(ω)(,x) = , ∀x, (.)

∂t

p(υ) –p(ω),q+ap(υ) –p(ω),q+υp(υ) –p(ω),q

=y(υ) –y(ω),q+ωυ,qp(ω), ∀qW,tJ, (.)

p(υ)(T,x) –p(ω)(T,x) = , ∀x. (.)

Inequalities (.) and (.) follow from the regularity estimates of (.)-(.) and

(.)-(.), respectively. Analogously, we can derive (.) and (.).

Lemma . Let(y,p,u)be the solution of(.)-(.).Assume that uL(H).We have

yh(Qhu) –yh(u)L(L)+yh(Qhu) –yh(u)L(H)Ch, (.)

ph(Qhu) –ph(u)L(L)+ph(Qhu) –ph(u)L(H)Ch. (.)

Proof It follows from (.) that

QhuuL(H–)ChuL(H). (.)

Settingυ=Qhuandω=uin (.)-(.), we obtain (.)-(.).

Lemma . Let(y(v),p(v))be the solution of(.)-(.)with vK.Suppose that y(v),p(v)∈

H(H).Then the following estimates hold:

Rhy(v) –yh(v)L(L)+Rhy(v) –yh(v)L(H)Ch, (.)

Rhp(v) –ph(v)L(L)+Rhp(v) –ph(v)L(H)Ch. (.)

Proof It follows from the definition ofRhand (.)-(.) that

∂t

y(v) –yh(v)

,wh

+aRhy(v) –yh(v),wh

+vy(v) –yh(v)

,wh

= ,

(7)

y(v)(,x) –yh(v)(,x) =y(x) –yh(x), ∀x, (.)

∂t

p(v) –ph(v)

,qh

+aqh,Rhp(v) –ph(v)

+vp(v) –ph(v)

,qh

=y(v) –yh(v),qh

, ∀qhWh,tJ, (.)

p(v)(T,x) –p(v)h(T,x) = , ∀x. (.)

Letwh=Rhy(v) –yh(v) in (.). From Hölder’s inequality, Young’s inequality with, and

(.) we derive

 

d

dtRhy(v) –yh(v) 

+cRhy(v) –yh(v)

 

∂t

Rhy(v) –yh(v)

,Rhy(v) –yh(v)

+aRhy(v) –yh(v),Rhy(v) –yh(v)

+vRhy(v) –yh(v)

,Rhy(v) –yh(v)

=∂t

Rhy(v) –y(v)

,Rhy(v) –yh(v)

+vRhy(v) –y(v)

,Rhy(v) –yh(v)

C()h∂ty(v)

 +C()h

y(v)

+ Rhy(v) –yh(v) 

. (.)

Note that

Rhy(v)(,x) –yh(v)(,x) = . (.)

Estimate (.) follows from (.) and Gronwall’s inequality. Similarly, we can obtain

(.).

4 A priori error estimates

In this section, we derive a priori error estimates of the approximation scheme (.)-(.). For ease of exposition, we set

J(u) =

T

yyd+u

dt,

Jh(uh) =

T

yhyd+uh

dt.

It is easy to show that

J(u),v=

T

(uyp,v)dt,

Jh(uh),v

=

T

(uhyhph,v)dt.

As in [], we assume that there exist neighborhoods of the exact solution uor of the

approximation solution uh inK and a constantc>  such that, for anyvorvh in this

neighborhood, the objective functional satisfies the following convexity conditions:

cvuL(L)

J(v) –J(u),vu, (.)

cvhuhL(L)

Jh(vh) –Jh(uh),vhuh

(8)

Theorem . Let(y,p,u)and(yh,ph,uh)be the solutions of(.)-(.)and(.)-(.).

Suppose that y,pH(L)L(H).Then

uuhL(L)Ch, (.)

yyhL∞(L)+yyhL(H)Ch, (.)

pphL∞(L)+pphL(H)Ch. (.)

Proof It follows from (.), (.), (.), and (.) that

cuuhL(L)

J(u) –J(uh),uuh

=

T

(uyp,uuh)dt

T

uhy(uh)p(uh),uuh

dtT

(yp,uuh) + (yhph,uhu)

+ (uhyhph,Qhuu) –

ypy(uh)p(uh),uuh

dt = T

(uhyhph,Qhuu) +

yhphy(uh)p(uh),uhu

dt = T

ypy(uh)p(uh),Qhuu

dt+

T

y(uh)p(uh) –yhph,Qhuu

dt

+

T

(yp,uQhu)dt+

T

yhphy(uh)p(uh),uhu

dt

:=I+I+I+I. (.)

For the first termI, by the embedding inequalityvL()CvH() and Young’s

in-equality withwe get

I=

T

ypy(uh)p(uh),Qhuu

dt

C()QhuuL(L)+Cy(u) –y(uh)

L(H)+Cp(u) –p(uh)

L(H). (.)

By Hölder’s inequality and the embedding inequalityvL()CvH()we have

I=

T

y(uh)p(uh) –yhph,Qhuu

dt

CQhuuL(L)+yhy(uh)

L(H)+php(uh)

L(H)

, (.)

and the third termIcan be estimates as follows:

I=

T

(yp,uQhu)dt

=

T

(ypQhyQhp,uQhu)dt

CyQhyL(H)+pQhpL(H)+QhuuL(L)

(9)

Applying Hölder’s inequality, the embedding inequalityvL()CvH(), and Young’s

inequality withtoI, we have

I=

T

yhphy(uh)p(uh),uhu

dt

C()yhy(uh)L(H)+php(uh)L(H)

+uuhL(L). (.)

According to (.), Lemmas .-., and (.)-(.), we obtain

uuhL(L)Ch. (.)

From (.)-(.) and (.)-(.) we have

∂t(yyh),wh

+a(yyh,wh) + (uyuhyh,wh) = , ∀whWh,tJ, (.)

y(,x) –yh(,x) =y(x) –Rhy(x), ∀x. (.)

Lettingwh=Rhyyhin (.), we get, for anytJ,

∂t(yyh),yyh

+a(yyh,yyh) +

u(yyh),yyh

=∂t(yyh),yRhy

+a(yyh,yRhy) +

yh(uhu),yyh

+u(yyh),yRhy

+yh(uuh),yRhy

. (.)

From (.), (.), (.), Young’s inequality with, and Gronwall’s inequality we derive

(.). It is paralleled to get (.).

5 Superconvergence analysis

In this section, we derive the superconvergence between projections of the exact solu-tions and approximation solusolu-tions. Let ube the solutions of (.)-(.). For a fixedt

(≤t∗≤T), we divideinto the following subsets:

+=τ:τ,a<ut∗,·<b

,

=τ:τ,ut∗,· τ =aorut∗,· τ =b

,

–=\+∪.

It is easy to see that these three subsets do not intersect with each other and=¯+∪

¯

∪ ¯–. We assume thatuandThare regular such thatmeas(–)≤Ch(see, e.g., []).

Theorem . Let(y,p,u)and(yh,ph,uh)be the solutions of(.)-(.)and(.)-(.),

respectively.Assume that all the conditions in Lemmas.-.are valid and y,pL(L∞).

Moreover,we suppose that the exact control,state,and costate solutions satisfy

(10)

Then,we have

QhuuhL(L)Ch

. (.)

Proof Lettingvh=Qhuin (.), we obtain the inequality

T

(uhyhph,Qhuuh)dt≥. (.)

It follows from the definition ofQh, (.), and (.) that

cQhuuhL(L)

Jh(Qhu) –Jh(uh),Qhuuh

=

T

Qhuyh(Qhu)ph(Qhu),Qhuuh

dt

T

(uhyhph,Qhuuh)dt

T

uyh(Qhu)ph(Qhu),Qhuuh

dt

=

T

(uyp,Qhuuh)dt+

T

yh(u)ph(u) –yh(Qhu)ph(Qhu),Qhuuh

dt

+

T

RhyRhpyh(u)ph(u),Qhuuh

dt+

T

(ypRhyRhp,Qhuuh)dt

=:I+I+I+I. (.)

For the first term, at timet∗(≤t∗≤T), we have

(uyp,Qhuuh) =

+

+

+

(uyp)(Qhuuh)dx (.)

and

(Qhuu)|= .

From (.) we get

(uyp)|+= .

Hence,

I=

T

–(uyp)(Qhuuh)dx dt

=

T

(uyp) –Qh(uyp)

(Qhuuh)dx dt

ChT

uyp,u,dt

ChT

uypW,∞uW,∞·measdt

ChuypL(W,∞)+uL(W,∞)

(11)

By using Hölder’s inequalty, the embedding inequalityvL()CvH(), and Young’s

inequality,IandIcan be estimated as follows:

I≤C()yh(u) –yh(Qhu)L(H)+ph(u) –ph(Qhu)L(H)

+QhuuhL(L) (.)

and

I≤C()Rhyyh(u)L(H)+Rhpph(u)L(H)

+QhuuhL(L). (.)

In addition, noting thaty,pL(L), we have

I≤C()

RhyyL(L)+RhppL(L)

+QhuuhL(L). (.)

It follows from (.)-(.), (.), and Lemmas .-. that (.) holds for small enough.

Theorem . Let(y,p,u)and(yh,ph,uh)be the solutions of(.)-(.)and(.)-(.),

respectively.Assume that all the conditions in Theorem.are valid.Then

yhRhyL∞(L)+yhRhyL(H)Ch

, (.)

phRhpL∞(L)+phRhpL(H)Ch

. (.)

Proof From the definition ofRh, (.)-(.), and (.)-(.), for anywhorqhWhand

tJ, we have

∂t(yhRhy),wh

+a(yhRhy,wh) +

u(yhRhy),wh

=∂t(yRhy),wh

+u(yRhy),wh

+yh(uQhu),wh

+yh(Qhuuh),wh

, (.)

yh(,x) –Rhy(,x) = , ∀x, (.)

∂t(phRhp),qh

+a(qh,phRhp) +

u(phRhp),qh

= –∂t(pRhp),qh

+u(pRhp),qh

+ph(uuh),qh

+ (yhRhy,qh), (.)

ph(T,x) –Rhp(T,x) = , ∀x. (.)

Hence, lettingwh=yhRhyin (.), (.) follows from (.)-(.), Hölder’s inequality,

Young’s inequality, Gronwall’s inequality, (.), and (.). Inequality (.) can be similarly

derived.

6 Numerical experiment

In this section, we present a numerical example to validate our superconvergence results. Lett> ,N=T/t∈Z+,t

n=nt,n= , , . . . ,N. Setϕn=ϕ(x,tn) and

dtϕn=

ϕnϕn–

(12)

By using the backward Euler scheme to approximate the time derivative, we introduce the following fully discrete approximation scheme: find (ynh,pnh–,uhn)∈Wh×Wh×Khsuch

that

dtynh,wh

+aynh,wh

+unhynh,wh

=fn,wh

, ∀whWh,n= , , . . . ,N, (.)

yh(x) =yh(x), ∀x, (.)

dtpnh,qh

+aqh,pnh–

+unhpnh–,qh

=ynhynd,qh

, ∀qhWh,n=N, . . . , , , (.)

pNh(x) = , ∀x, (.)

uhnynhphn–,vhunh

≥, ∀vhKh,n= , , . . . ,N. (.)

Let= (, )×(, ),T = ,a= ,b= ., andA(x) be a unit matrix. The following example is solved numerically by a precondition projection algorithm (see e.g. []), where the codes are developed based on AFEPack, which is freely available.

Example  The data are as follows:

p(t,x) =sin(πx)sin(πx)( –t),

y(t,x) =sin(πx)sin(πx)t,

u(t,x) =min.,max,y(t,x)p(x,t),

f(t,x) =yt(t,x) –div

A(x)∇y(t,x)+u(t,x)y(t,x),

yd(t,x) =y(t,x) +pt(t,x) +div

A∗(x)∇p(t,x)–p(t,x)y(t,x).

For brevity, we set

|||ϕ|||=

Nl

n=–l

tϕnL()

 

and

|||ϕ|||=

Nl

n=–l

tϕnH()

 

,

wherel=  for the controluand the statey, andl=  for the costatep. In Table , the errors |||Qhuuh|||,|||Rhyyh|||, and|||Rhpph|||on a sequence of uniformly refined meshes are

listed. It is consistent with our superconvergence results in Section .

Whenh= .e–,t= , andt= ., we plot the profile ofuhin Figure .

7 Conclusions

(13)
[image:13.595.117.478.93.364.2]

Table 1 The errors of Example 1

t h |||Qhu – uh||| |||Rhy – yh|||1 |||Rhp – ph|||1

1/10 1.0e–1 9.94516e–2 2.68512e–2 4.72410e–2 1/30 5.0e–2 3.41740e–2 8.46440e–3 1.47458e–2 1/90 2.5e–2 1.20365e–2 2.78014e–3 4.68123e–3 1/270 1.25e–2 3.58914e–3 8.83804e–4 1.50701e–3

Figure 1The numerical solutionuhatt= 0.5 in Example 1.

optimal control problems (see, e.g., [–]). Although bilinear optimal control problems are frequently met in applications, they are much more difficult to handle in comparison to linear or semilinear cases. There is little work on bilinear optimal control problems. Recently, Yang et al. [] investigated a priori error estimates and superconvergence of fi-nite element methods for bilinear elliptic optimal control problems. Hence, our results on bilinear parabolic optimal control problems are new.

Competing interests

Both authors declare that they have no competing interests.

Authors’ contributions

The authors have participated in the sequence alignment and drafted the manuscript. They have approved the final manuscript.

Acknowledgements

The first author is supported by the National Natural Science Foundation of China (11401201), the Hunan Province Education Department (16B105), and the construct program of the key discipline in Hunan University of Science and Engineering. The second author is supported by the scientific research program in Hunan University of Science and Engineering (2015).

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 9 September 2016 Accepted: 8 March 2017 References

1. Liu, W, Yan, N: Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Science Press, Beijing (2008) 2. Arada, N, Casas, E, Tröltzsch, F: Error estimates for semilinear elliptic control problem. Comput. Optim. Appl.23,

201-229 (2002)

(14)

4. Chen, Y, Dai, Y: Superconvergence for optimal control problems governed by semi-linear elliptic equations. J. Sci. Comput.39, 206-221 (2009)

5. Yang, D, Chang, Y, Liu, W: A priori error estimate and superconvergence analysis for an optimal control problem of bilinear type. J. Comput. Math.26(4), 471-487 (2008)

6. Chen, Y, Huang, Y, Liu, W, Yan, N: Error estimates and superconvergence of mixed finite element methods for convex optimal control problems. J. Sci. Comput.42(3), 382-403 (2010)

7. Chen, Y, Hou, T, Zheng, W: Error estimates and superconvergence of mixed finite element methods for optimal control problems with low regularity. Adv. Appl. Math. Mech.4(6), 751-768 (2012)

8. Meidner, D, Vexler, B: A priori error estimates for space-time finite element discretization of parabolic optimal control problems. Part II: problems with control constraints. SIAM J. Control Optim.47(3), 1301-1329 (2008)

9. Tang, Y, Hua, Y: Superconvergence analysis for parabolic optimal control problems. Calcolo51, 381-392 (2014) 10. Tang, Y, Chen, Y: Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal

control problems. Front. Math. China8(2), 443-464 (2013)

11. Dai, Y, Chen, Y: Superconvergence for general convex optimal control problems governed by semilinear parabolic equations. ISRN Appl. Math.2014, Article ID 579047 (2014)

12. Lions, J, Magenes, E: Non-homogeneous Boundary Value Problems and Applications. Springer, Berlin (1972) 13. Lions, J: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

14. Liu, W, Yan, N: A posteriori error estimates for optimal control problems governed by parabolic equations. Numer. Math.93, 497-521 (2003)

Figure

Table 1 The errors of Example 1

References

Related documents

Even although we know general Cloud usage can be beneficial, we do not know if this translates to the work environment when using business focused net-centric systems.. Is it the

This shows that the method is very accurate for the determination of the Atovaquone in Various pharmaceutical dosage forms ( Fig... The assay results shows that

Based on the test results, it was observed that a maximum of 11.38 %, 7.87 % and 10.67 % increase in compressive strength, split tensile strength and

A new series of 2-(4-aminophenyl) benzothiazole derivatives with a cyano or alkynyl group at 3’ position was prepared and evaluated for antitumor activity..

(These cases were selected in order to be representative of the larger group of all participants with visual impair- ment in terms of the degree of visual impairment, age, sex

We make 3 key observations with respect to this: (i) since the preservation of language semantics is necessary for transformation, whereas only the attribute or style of the text

Pritom [8] applied Naive Bayes, C4.5 Decision Tree and Support Vector Machine (SVM) classification algorithms to calculate the prediction accuracy of breast

In this work, to experimentally analyze the performance of purposed algorithm we first randomly deployed the sensor nodes in the region of area 200*200m. We also randomly